# Does the ruling out of TeV scale SUSY breaking disfavor grand unification?

One of the arguments in favor of TeV scale SUSY breaking is that it leads to the appropriate running of the gauge coupling strengths leading to grand unification, i.e. $k_Y = \frac{5}{3}$ instead of $k_Y = \frac{4}{3}$. With the LHC ruling out TeV scale SUSY breaking, what is the current consensus on grand unification? I know it's always possible to restore grand unification if you really insist upon it with contrived mechanisms like split SUSY, brane worlds with fine-tuned couplings, exotic fields, etc. but what would you say the current Bayesian posterior probability for grand unification is right now? How much would you be willing to bet on it? Isn't it kind of suspicious we've never detected proton decay or magnetic monopoles so far? The doublet-triplet splitting problem also makes you wonder...

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Did the LHC rule out SUSY breaking at TeV already? I think this is just a preliminary guess. –  Ron Maimon Sep 24 '11 at 2:17

### Wagers

You asked for opinions and wagers, so I'll give it, although these things are not worth the electrons they are written on. I believe grand unification is a pretty good bet with or without TeV scale SUSY, and I would take a bet at 9:1 odds that it is really there. It is not plausible that the hypercharge assignments match such simple representations of SU(5), or one representation of SO(10), in such a natural way. If there is no GUT, you could add random reps that don't fit nicely into any GUT, and this is the vast majority of low-energy Lagrangians. Even with SU(5) the doublet Higgs doesn't fit naturally (the only thing), but we don't know the Higgs sector fully yet, so this is not much of a concern.

TeV scale SUSY is not ruled out at all yet, alhough I would only put even money on the thing at best, and I am not sure which side I would take on the bet, personally, because MSSM is such a contrived mess. If there were some simplification gained, like the Higgs a superpartner, then I would take a 2:1 wager, but there isn't, and some standard model stuff is wrecked. MSSM is mostly useful because it shows you how to do some simple SUSY string compactifications and get something like the standard model out, it doesn't feel compelling as physics.

### Nonrenormalizable terms

Anyway, that's subjective crap. Here's the main thing. The coupling unification prediction is not a good test by itself, and the observed SUSY matching of couplings is a little too good in SUSY GUTs. One should expect a certain failure of the coupling unification from non-renormalizable terms at the GUT scale, and that this failure is absent in the SUSY models is a little suspicious.

The field theory predictions for unification assume there are no non-renormalizable terms. The GUT scale is always close enough to the Planck scale to make this assumption implausible. For the simplest example, consider the standard SU(5) gut with a breaking to the standard model through the standard SU(5) adjoint Higgs field $H$ which has VEV $\mathrm{diag}(A,A,A,B,B)$. In this theory, introduce the natural dimension 5 operator:

$${\alpha \over M} tr H G G$$

This gives couplings of SU(2) and SU(3) modified by

$$\Delta {1\over g_3^2} \propto \alpha {A\over M}$$ $$\Delta {1\over g_2^2} \propto \alpha {B\over M}$$ $$\Delta {1\over g_1^2} \propto \alpha {2A+3B\over 5M}$$

Where the proportionality constant is the same for all 3, depending only on the generator inner product convention. if you define the GUT coupling to be the strong SU(3) coupling, the difference in the other couplings at the GUT scale is

$$\Delta {1\over g_2^2} = \alpha {B-A\over M}$$ $$\Delta {1\over g_1^2} = \alpha {3(B-A)\over 5M}$$

So the U(1) coupling is shifted by 3/5 the amount the SU(2) coupling is shifted. This is enough to determine the possible points where SU(5) unification can happen--- you look for a point where the SU(3)-SU(2) difference is 5/3 the SU(3)-U(1) difference. Looking at the coupling graph, the points are too low in energy for proton decay, they are around $10^{15}$ GeV, so this type of fix is ruled out.

Using dimension 6 terms, you can get whatever you want for couplings and unification, and if the unification is at $10^{17}$ GeV, you might expect hefty dimension 6 contributions. But if you have dimension 6 terms important at the GUT scale, you should probably be doing string theory to get the coefficients, not piddling around with effective actions. I think to call a theory a GUT, you want to include leading corrections only.

The fact that the natural dimension 5 fix doesn't work suggests that it doesn't work in SO(10) either, although you have to list out all the dimension 5 operators there. You can also try very high order terms (like using SU(5) epsilon tensor) to reproduce miniscule things, there is no analysis of the effect of nonrenormalizable corrections, and there probably is a host of models which work fine.

The failure of couplings we see is not inconsistent with the level of coupling modification expected from a dimension 5 term. I asked about this here a few months ago: Non-renormalizable corrections to GUT unification.

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